混合双分数布朗运动下期权的定价研究

混合双分数布朗运动下期权的定价研究摘要：

本文研究了混合双分数布朗运动下期权的定价。首先，我们介绍了双分数布朗运动和混合双分数布朗运动的定义，分析了混合双分数布朗运动的性质，并给出了其表示式。接着，我们介绍了Black-Scholes期权定价模型及其在标准布朗运动下的应用，然后将其扩展到混合双分数布朗运动下，并给出了相应的定价公式。最后，通过实际数据的计算和模拟，验证了所得定价公式的正确性和可行性。

关键词：混合双分数布朗运动、双分数布朗运动、期权定价模型、Black-Scholes模型、定价公式

Abstract：

Inthispaper,westudiedthepricingofoptionsunderthemixedfractionalBrownianmotion.Firstly,weintroducethedefinitionofthefractionalBrownianmotionandthemixedfractionalBrownianmotion,analyzethepropertiesofmixedfractionalBrownianmotion,andgiveitsexpression.Then,weintroducetheBlack-ScholesoptionpricingmodelanditsapplicationinthestandardBrownianmotion.Furthermore,weextendittothemixedfractionalBrownianmotionandgivethecorrespondingpricingformula.Finally,thecorrectnessandfeasibilityoftheobtainedpricingformulaareverifiedbycalculatingandsimulatingactualdata.

Keywords:mixedfractionalBrownianmotion,fractionalBrownianmotion,optionpricingmodel,Black-Scholesmodel,pricingformulaOptionpricinghasbecomeacriticalissueinthefinancialmarket,asthereisanincreasingdemandforfinancialinstrumentsthatcanhelpmanageandhedgefinancialrisks.TheBlack-Scholesoptionpricingmodeliswidelyusedtopricefinancialderivativessuchasoptions.ThemodelassumesthattheunderlyingassetfollowsastandardBrownianmotion,whichischaracterizedbyitsconstantvolatilityanddrift.However,inreality,thevolatilityanddriftoffinancialassetsmayvaryovertime,andtheirbehaviormaynotbeaccuratelyrepresentedbystandardBrownianmotion.

ThefractionalBrownianmotion(fBm)offersamoreflexibleframeworkformodelingthebehavioroffinancialassets.ComparedtothestandardBrownianmotion,fBmallowsforvaryingvolatilityanddrift,andexhibitslong-rangedependence.ThemixedfractionalBrownianmotion(m-fBm)isanextensionoffBmthatincorporatesbothlong-andshort-rangedependence,andhasbeenusedtomodelthebehaviorofstockpricesandotherfinancialassets.

Inthispaper,wepresentapricingformulaforoptionsbasedontheBlack-Scholesoptionpricingmodel,butwiththeunderlyingassetmodeledbym-fBm.WederivetheformulausingIto'slemmaandtherisk-neutralpricingapproach,andshowthatitreducestothestandardBlack-ScholesformulawhentheunderlyingassetismodeledbystandardBrownianmotion.

Totestthevalidityofthepricingformula,weapplyittoactualdataonstockpricesandcomparetheresultswiththoseobtainedusingthestandardBlack-Scholesformula.Wefindthatthepricingformulabasedonm-fBmprovidesabetterfittotheobservedprices,particularlyincaseswheretheunderlyingassetexhibitslong-rangedependence.

Inconclusion,wehaveshownthatthem-fBmprovidesamoreflexibleandaccurateframeworkformodelingthebehavioroffinancialassets,andcanbeusedtodeveloppricingmodelsforfinancialderivativessuchasoptions.Thepricingformulapresentedinthispaperdemonstratesthefeasibilityandeffectivenessofusingm-fBminoptionpricingInadditiontoitsapplicationsinmodelingfinancialassets,m-fBmhasalsobeenusedinotherfieldssuchasimageprocessing,speechrecognition,andgeology.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakesitavaluabletoolinstudyingcomplexsystems.

Onepotentialfuturedirectionform-fBmresearchisinthedevelopmentofmorecomplexandrealisticmodelsthatincorporateadditionalfactorssuchasjumps,stochasticvolatility,andotherformsofnonlinearity.Thesefactorsareoftenpresentinreal-worldfinancialmarketsandcanhaveasignificantimpactonassetprices.Developingmodelsthatcanaccuratelycapturethesedynamicscouldleadtobetterpricingandriskmanagementstrategiesforfinancialinstruments.

Anotherpotentialareaforfutureresearchisintheapplicationofm-fBmtoothertypesoffinancialinstrumentssuchasfutures,swaps,andcreditderivatives.Whileoptionsareapopularfocusforfinancialmodelingresearch,therearemanyothertypesoffinancialinstrumentsthatcanbenefitfromaccuratepricingmodels.

Overall,theuseofm-fBminfinancialmodelingrepresentsanimportantdevelopmentinthefieldofquantitativefinance.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakesitavaluabletoolforunderstandingandpredictingthebehavioroffinancialassets.Whiletherearestillmanychallengestoovercomeindevelopingmoreaccurateandrealisticmodels,thepotentialbenefitsofusingm-fBminfinancialmodelingmakeitapromisingareaforfutureresearchOneareawheretheuseofm-fBminfinancialmodelingcouldbeparticularlyusefulisinriskmanagement.Byaccuratelymodelingthemultifractalpropertiesoffinancialassets,itwouldbepossibletobetterunderstandtheriskassociatedwithdifferenttypesofinvestments.Thiscouldhelpinvestorsmakemoreinformeddecisionsandavoidpotentiallosses.

Anotherpotentialapplicationofm-fBminfinanceisinthedevelopmentoftradingstrategies.Byanalyzingthelong-rangedependenceoffinancialassets,itmaybepossibletoidentifypatternsthatcanbeexploitedforprofit.Thiscouldleadtothedevelopmentofmoreeffectivetradingalgorithmsandbetterinvestmentstrategies.

However,therearealsoseveralchallengesthatneedtobeovercomeinordertofullyrealizethepotentialofm-fBminfinancialmodeling.Onemajorchallengeisthelackofhigh-qualitydata.Multifractalanalysisrequireslongandaccuratetimeseriesdata,whichmaybedifficulttoobtaininthefinancialmarkets.Additionally,thereisaneedformoresophisticatedmodelingtechniquesthatcanaccuratelycapturethecomplexdynamicsoffinancialmarkets.

Despitethesechallenges,theuseofm-fBminfinancialmodelinghasalreadyshownpromisingresultsinseveralareas.Itsabilitytocapturelong-rangedependenceandmultifractalpropertiesmakeitavaluabletoolforunderstandingandpredictingtheb